Lecture 16-2007 - Lecture XVI The characteristic function...

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Unformatted text preview: Lecture XVI The characteristic function of a random variable X is defined as Note that this definition parallels the definition of the moment-generating function ( 29 ( 29 ( 29 ( 29 ( 29 cos sin cos sin itX X t E e E tX i tX E tX iE tX = = + = + ( 29 tX X M t E e = Like the moment-generating function there is a one-to-one correspondence between the characteristic function and the distribution of random variable. Two random variables with the same characteristic function are distributed the same. The characteristic function of the uniform distribution function of the uniform distribution function is The characteristic function of the Normal distribution function is ( 29 1 it X t e =- ( 29 2 2 2 t it X t e - = The Gamma distribution function which implies the characteristic function ( 29 ( 29 ( 29 1 0, r r X x e f X X r -- = = ( 29 ( 29 1 1 X r t it =- Taking a Taylor series expansion of around the...
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This note was uploaded on 11/08/2011 for the course AEB 6933 taught by Professor Carriker during the Fall '09 term at University of Florida.

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Lecture 16-2007 - Lecture XVI The characteristic function...

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